🧮 algebra

1. **State the problem:** Simplify the expression $$\frac{x^2 + 3x}{x + 10} - \frac{3x + 100}{x + 10}$$. 2. **Identify the formula and rules:** Since both fractions have the same d

1. **State the problem:** A teacher gave each student 4 purple cards and 3 white cards. The total number of cards given out is 91. We need to find:

1. **State the problem:** We know the elevator travels 330 feet in 10 seconds. We want to find how far it travels in 11 seconds. 2. **Identify the formula:** Since the elevator mov

1. **Problem Statement:** Prove by mathematical induction that for all positive integers $n$:

1. We are asked to solve the inequality $$2 < \frac{2x + 3}{x + 1} < \frac{5}{2}$$ for real numbers $x$ such that $x > 1$. 2. Important: Since $x > 1$, the denominator $x + 1 > 0$,

1. The problem is to find the value of the function $y = 2x^2 - 3x + 5$ when $x = 4$. 2. The formula given is a quadratic function: $$y = 2x^2 - 3x + 5$$ where $x$ is the input var

1. **State the problem:** Simplify the expression $$\frac{x^2 - 36}{x + 4} \cdot \frac{4x}{6x - 36}$$. 2. **Recall important formulas and rules:**

1. **Problem statement:** Simplify the expression $2x + 2 - 3x - 3 + 6x^2 - 6$. 2. **Formula and rules:** Combine like terms by adding or subtracting coefficients of the same power

1. **State the problem:** Evaluate the expression $$\sqrt[3]{3k} + 5k$$ when $$k = -72$$. 2. **Recall the formula and rules:** The cube root of a number $$a$$ is a value $$b$$ such

1. The problem is to understand and analyze the function $y = x^2$. 2. This is a quadratic function, which generally has the form $y = ax^2 + bx + c$. Here, $a=1$, $b=0$, and $c=0$

1. **State the problem:** Simplify the expression $$\frac{5x}{2y} \div \frac{3x}{4y}$$. 2. **Recall the rule for division of fractions:** Dividing by a fraction is the same as mult

1. The problem is to understand and express the general form of the $n$th root of $x$, written as $\sqrt[n]{x}$. 2. The $n$th root of a number $x$ is the number that, when raised t

1. **State the problem:** Evaluate the expression $5x^2 + 2x + 6$ when $x=3$. 2. **Recall the formula:** The expression is a quadratic polynomial in $x$.

1. **State the problem:** We have an arithmetic sequence $2, 4, 6, 8, \ldots$ and want to find how many terms sum to 600. 2. **Recall the formula for the sum of an arithmetic seque

1. **Énoncé du problème :** Montrer plusieurs propriétés sur les sous-groupes $H$, $K$ de $G$ avec $H$ normal dans $G$. 2. **Définition et rappel :** $HK = \{hk \mid h \in H, k \in

1. **State the problem:** Simplify the expression $$\frac{x - 2}{x^2 - 16} = x(x + 4)(x - 4) - x(x + 4)(x - 4) \over x^2 - 4x$$. 2. **Rewrite the expression:** Notice the right sid

1. Zadatak: Riješiti matričnu jednadžbu $$X \cdot A - 2B = X$$ gdje su $$A = \begin{bmatrix}2 & -1 \\ 5 & 3 \end{bmatrix}, \quad B = \begin{bmatrix}0 & -1 \\ 3 & 5 \end{bmatrix}.$$

1. Calculer et simplifier : - $$\sqrt{12} \times \sqrt{3} = \sqrt{12 \times 3} = \sqrt{36} = 6$$

1. Planteamos el problema: Dada la función $$f(x) = \log_4(x+2) + \log_4(x-2) - \log_4 5,$$ queremos encontrar para qué valores de $x$ se cumple que $f(x) < 0$. 2. Recordemos que e

1. **Problem statement:** Find a rational function $f$ defined on its maximum domain that has a vertical asymptote at $x=7$. 2. **Recall:** A vertical asymptote occurs where the de

1. **Stating the problem:** We need to show that the determinant of the matrix $$\begin{vmatrix}-a^2 & 2ab & ac \\ ba & -b^2 & bc \\ ca & cb & -c^2\end{vmatrix} = 4a^2b^2c^2.$$